How do you show that #f(x)=6x^2  24x + 22# satisfies the hypotheses of Rolle's theorem on [0,4]?
Use the fact that this function is a polynomial.
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To show that ( f(x) = 6x^2  24x + 22 ) satisfies the hypotheses of Rolle's theorem on the interval ([0,4]), we need to demonstrate the following:
 The function ( f(x) ) is continuous on ([0,4]).
 The function ( f(x) ) is differentiable on ( (0,4) ).
 ( f(0) = f(4) ).
Let's verify each condition:

Since ( f(x) = 6x^2  24x + 22 ) is a polynomial function, it is continuous everywhere, including on the interval ([0,4]).

( f(x) = 6x^2  24x + 22 ) is a polynomial function, which implies it is differentiable everywhere. Therefore, it is differentiable on the interval ( (0,4) ).

( f(0) = 22 ) and ( f(4) = 6(4)^2  24(4) + 22 = 22 ).
Since all three conditions are satisfied, we can conclude that ( f(x) = 6x^2  24x + 22 ) satisfies the hypotheses of Rolle's theorem on the interval ([0,4]).
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When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
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